Multiple periodic solutions of second order parameter-dependent equations via rotation numbers

Chunlian Liu; Shuang Wang; Chunlian Liu; Shuang Wang

doi:10.3934/math.20231285

AIMS Mathematics

2023, Volume 8, Issue 10: 25195-25219. doi: 10.3934/math.20231285

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Research article

Multiple periodic solutions of second order parameter-dependent equations via rotation numbers

Chunlian Liu ¹,
Shuang Wang ^{2
,
,}

1.
School of Science, Nantong University, Nantong 226019, China
2.
School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224051, China

Received: 06 June 2023 Revised: 11 August 2023 Accepted: 24 August 2023 Published: 29 August 2023
MSC : 34C25, 34C15, 34B15

We investigate the existence of multiple periodic solutions for a class of second order parameter-dependent equations of the form $ x''+f(t, x) = sp(t) $. We compare the behavior of its solutions with suitable linear and piecewise linear equations near positive infinity and infinity. Furthermore, in this context, the nonlinearity $ f $ does not satisfy the usual sign condition, and the global existence of solutions for the Cauchy problem associated to the equation is not guaranteed. Our approach is based on the Poincaré-Birkhoff twist theorem, a rotation number approach and the phase-plane analysis. Our result generalizes the result in Fonda and Ghirardelli ^[1] for second order parameter-dependent equations.
- periodic solutions,
- indefinite term,
- Poincaré-Birkhoff twist theorem,
- rotation number,
- parameter-dependent
Citation: Chunlian Liu, Shuang Wang. Multiple periodic solutions of second order parameter-dependent equations via rotation numbers[J]. AIMS Mathematics, 2023, 8(10): 25195-25219. doi: 10.3934/math.20231285

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Abstract

We investigate the existence of multiple periodic solutions for a class of second order parameter-dependent equations of the form $ x''+f(t, x) = sp(t) $. We compare the behavior of its solutions with suitable linear and piecewise linear equations near positive infinity and infinity. Furthermore, in this context, the nonlinearity $ f $ does not satisfy the usual sign condition, and the global existence of solutions for the Cauchy problem associated to the equation is not guaranteed. Our approach is based on the Poincaré-Birkhoff twist theorem, a rotation number approach and the phase-plane analysis. Our result generalizes the result in Fonda and Ghirardelli ^[1] for second order parameter-dependent equations.

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